| L(s) = 1 | + (−1.36 − 0.387i)2-s + (0.794 + 0.362i)3-s + (1.69 + 1.05i)4-s + (−0.649 + 2.21i)5-s + (−0.939 − 0.801i)6-s + (1.59 − 4.60i)7-s + (−1.90 − 2.09i)8-s + (−1.46 − 1.69i)9-s + (1.74 − 2.75i)10-s + (−1.51 + 0.368i)11-s + (0.967 + 1.45i)12-s + (−1.76 − 3.41i)13-s + (−3.95 + 5.64i)14-s + (−1.31 + 1.52i)15-s + (1.77 + 3.58i)16-s + (−6.54 − 2.62i)17-s + ⋯ |
| L(s) = 1 | + (−0.961 − 0.274i)2-s + (0.458 + 0.209i)3-s + (0.849 + 0.527i)4-s + (−0.290 + 0.989i)5-s + (−0.383 − 0.327i)6-s + (0.602 − 1.74i)7-s + (−0.672 − 0.740i)8-s + (−0.488 − 0.563i)9-s + (0.550 − 0.871i)10-s + (−0.457 + 0.110i)11-s + (0.279 + 0.419i)12-s + (−0.488 − 0.947i)13-s + (−1.05 + 1.50i)14-s + (−0.340 + 0.392i)15-s + (0.443 + 0.896i)16-s + (−1.58 − 0.635i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 536 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.144 + 0.989i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 536 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.144 + 0.989i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.505257 - 0.584532i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.505257 - 0.584532i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (1.36 + 0.387i)T \) |
| 67 | \( 1 + (1.02 - 8.12i)T \) |
| good | 3 | \( 1 + (-0.794 - 0.362i)T + (1.96 + 2.26i)T^{2} \) |
| 5 | \( 1 + (0.649 - 2.21i)T + (-4.20 - 2.70i)T^{2} \) |
| 7 | \( 1 + (-1.59 + 4.60i)T + (-5.50 - 4.32i)T^{2} \) |
| 11 | \( 1 + (1.51 - 0.368i)T + (9.77 - 5.04i)T^{2} \) |
| 13 | \( 1 + (1.76 + 3.41i)T + (-7.54 + 10.5i)T^{2} \) |
| 17 | \( 1 + (6.54 + 2.62i)T + (12.3 + 11.7i)T^{2} \) |
| 19 | \( 1 + (-3.13 + 1.08i)T + (14.9 - 11.7i)T^{2} \) |
| 23 | \( 1 + (1.62 + 2.28i)T + (-7.52 + 21.7i)T^{2} \) |
| 29 | \( 1 + (-4.51 + 2.60i)T + (14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 + (-6.32 - 3.25i)T + (17.9 + 25.2i)T^{2} \) |
| 37 | \( 1 + (0.358 + 0.207i)T + (18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (-1.38 + 1.09i)T + (9.66 - 39.8i)T^{2} \) |
| 43 | \( 1 + (11.2 - 1.61i)T + (41.2 - 12.1i)T^{2} \) |
| 47 | \( 1 + (0.299 - 0.0286i)T + (46.1 - 8.89i)T^{2} \) |
| 53 | \( 1 + (-6.69 - 0.962i)T + (50.8 + 14.9i)T^{2} \) |
| 59 | \( 1 + (-0.460 - 0.717i)T + (-24.5 + 53.6i)T^{2} \) |
| 61 | \( 1 + (-6.83 - 1.65i)T + (54.2 + 27.9i)T^{2} \) |
| 71 | \( 1 + (-1.19 + 0.479i)T + (51.3 - 48.9i)T^{2} \) |
| 73 | \( 1 + (-1.94 + 8.00i)T + (-64.8 - 33.4i)T^{2} \) |
| 79 | \( 1 + (-0.643 + 13.4i)T + (-78.6 - 7.50i)T^{2} \) |
| 83 | \( 1 + (8.06 + 8.45i)T + (-3.94 + 82.9i)T^{2} \) |
| 89 | \( 1 + (-3.42 - 7.49i)T + (-58.2 + 67.2i)T^{2} \) |
| 97 | \( 1 + (-1.45 + 2.51i)T + (-48.5 - 84.0i)T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−10.31231750765744793836359658345, −10.09082284524489817080619564701, −8.739393567434284871426889993795, −7.928900666258706142993687633640, −7.18316195970427107888487454178, −6.54731710592358153048759788002, −4.61987689161411669895087294085, −3.40250209520289140786777490611, −2.59001148553607481391802273912, −0.55562196723417641480669330965,
1.82527810047737144706565725552, 2.59724842473809967202773993304, 4.80012364497896578566948931165, 5.51396582179196310011823958299, 6.66353801523654530446620573955, 8.010766325956654239360357061325, 8.495293299763514139729071395179, 8.899171101126402102517541084858, 9.849012791687060434421195121963, 11.22587804270482416517843712009
